AN INTRODUCTION TO STABILITY CONDITIONS FRANCO ROTA Abstract. In these notes we introduce the notion of stability condition on the category of coherent sheaves on a smooth projective variety. These notes are meant to be a guide for someone approaching the subject for the first time, they are focused on examples and motivation to help intuition. Contents 1. Introduction and motivation1 2. Stability for vector bundles on curves3 3. Stability conditions5 4. Examples6 4.1. Numerical stability conditions on curves of positive genus6 4.2. A first approach at surfaces6 4.3. Perspectives7 References7 1. Introduction and motivation The classification problem which permeates many areas of study in algebraic geometry is probably the main motivation for the study of moduli spaces. The ideas behind a moduli space are very simple, and represent attempts to answer questions like: is there a way to parametrize the set of objects I'm interested in? How far are these two objects from being isomorphic to each other? What does "far" in the previous question even mean? One encounters moduli spaces very soon in algebraic geometry, often without even noticing. A simple example of a moduli space is a linear series: we can regard n jOP (k)j as a projective space whose points represent a hypersurface of degree k in Pn. In some sense, a moduli space is a geometric object whose points represent isomorphism classes of elements of some set. To be more precise, let's focus on moduli spaces of coherent sheaves on a variety. Given a smooth projective variety X over a field k, we want to construct a scheme which parametrizes isomorphism classes of coherent sheaves on X. The case of line bundles is well understood: there exists a scheme Pic X , called the Picard scheme of X, whose k-rational points form the Picard group Pic (X). This scheme is neither projective nor finite type, but it decomposes as G P Pic X = Pic X and each component is projective and parametrizes line bundles with a given Hilbert polynomial with respect to a very ample class O(1). The Hilbert polynomial of 1 2 FRANCO ROTA a line bundle L is χ(X; L(m)) = R ch(L)ch(O(m))td(S) (recall the Hirzebruch- Riemann-Roch formula). It only depends on the choice of the ample class and c1(L). This suggests that when we try to generalize to higher rank we will need to make assumptions on numerical invariants, if we want to obtain components which are reasonably well-behaved. Unfortunately, fixing the Hilbert polynomial won't be enough to get the nice geometric properties we are interested in, as illustrated by the next examples. Example 1.1. Consider the family of rank 2 vector bundles En = O(n) ⊕ O(−n) on P1. These are not isomorphic (for example, their spaces of global sections have 0 different dimensions, h (En) = n + 1). However, their Hilbert polynomials agree: ch(En) = rkEn + c1(En) = (2; 0) for all En. If a moduli space representing in particular the En existed, even indexing its components with the Hilbert polynomial 0 wouldn't give a scheme of finite type: the condition h (En) ≥ m is closed, so we would get an infinite, stricly descending chain of closed subschemes. 1 1 1 1 1 Example 1.2. On P the group Ext (O(1); O(−1)) = H (P ;!P ) has dimension one. This means that non-trivial extensions 0 !O(−1) ! Eλ !O(1) ! 0 are isomorphic to O ⊕O (the only other rank 2 vector bundle with 2 global sections and trivial first Chern class). This yields a rank 2 vector bundle over A1 × P1 resctricting to O ⊕ O above λ × P1 for λ 6= 0 and to O(−1) ⊕ O(1) above zero. Again, assume that a moduli space M existed. By its universal property, the vector bundle above corresponds to a map A1 ! M which sends the line minus the origin in a point x corresponding to O ⊕ O and the origin to a different point y. This gives a way to extend the constant map to x defined on A1 − 0 which is different from the obvious assignment 0 7! x. In other words, M is not a separated scheme. It is evident at this point that we need to add some extra condition to hope for the existence a well-behaved moduli space. To make a more precise statement, if X is a smooth projective variety over k, and P is a fixed Hilbert polynomial, denote by (∗) the aforementioned extra condition. We are interested in the functor M :(Sch=k) ! Sets S 7! fE 2 Coh(S × X)jE S-flat, 8s 2 S : P (Es) = P; Es satisfies (∗)g = ∼ where the equivalence relation is given by E ∼ E ⊗ p∗M for some M 2 Pic (S). Definition 1.3. We say that a functor M is corepresented by a scheme M if there exists a natural transformation α : M! Hom(−;M) such that every other transformation M! Hom(−;N) factors through a unique M ! N, i.e.: α M Hom(−;M) 9! Hom(−;N) In this case, M is said to be a moduli space for M. It is a coarse moduli space if we have a bijection M(k) ! M(k). It is a fine moduli space if α is an isomorphism. In this case we say that M represents M. This last condition is equivalent to the existence of a universal family over M × X AN INTRODUCTION TO STABILITY CONDITIONS 3 Our hope is to find a notion of stability which can guarantee the existence of a moduli space for M. In these notes, rather than explaining why an efficient notion of stability is the one presented below, we want to illustrate how this notion generalizes to a much broader framework. 2. Stability for vector bundles on curves Consider a smooth projective curve C over an algebraically closed field, and denote by Coh(C) the cateogry of coherent sheaves on C. Definition 2.1. A coherent sheaf E 2 Coh(C) is said to be µ-stable (resp. µ- semistable) if E is torsion free (i.e. locally free), and all proper torsion free sub- sheaves 0 6= F ⊂ E satisfy µ(F ) < µ(E) (resp. µ(F ) ≤ µ(E)) deg(E) where µ(E) = is called the slope of E. rk(E) The notion of µ-stability, or the more general definition of stability `ala Gieseker, turns out to be a suitable notion of stability for the construction of a moduli space of sheaves, indeed: Theorem 2.2. Fix a smooth projective curve C, and a Hilbert polynomial P . Then the functor M :(Sch=k) ! Sets S 7! fE 2 Coh(S × C)jE S-flat, 8s 2 S : P (Es) = P; Es µ-stable g = ∼iso is corepresented by a projective scheme M. The closed points of M parametrize isomorphism classes of stable sheaves with Hilbert polynomial P .1 We shall rewrite stability in a way which is better suited to the generalizations we want to make. Define Z(E) = −deg(E) + irk(E) and let the phase φ(E) 2 (0; 1] of a nonzero sheaf E be uniquely defined by Z(E) 2 exp(iπφ(E)) · R>0 Then, the stability function Z defines Z : Coh(C) − f0g ! H := H [ R<0: We can talk about phases rather than slopes, because of the following: Lemma/Definition 2.3. Suppose E 2 Coh(C) is locally free. Then E is µ-stable iff for all proper subsheaves 0 6= F ⊂ E we have the following inequality: (1) φ(F ) < φ(E): Likewise for µ-semistability. Moreover, observe the formulae µ(E) = − cot(πφ(E)) and πφ(E) = cot−1(−µ(E)). Z(E) Proof. Write = −µ(E) + i. rk(E) 1For more on this part, see [2]. 4 FRANCO ROTA It seems natural to extend the definition of (semi)stability to the whole Coh(C) in terms of the (weak) inequality (1). This allows to treat torsion sheaves and vector bundles with the same machinery. The following properties will guide us in generalizing the notion of stability. Proposition 2.4. The following semi-orthogonality properties hold: • Let 0 ! K ! E ! G ! 0 be a short exact sequece in Coh(C). Then φ(K) < φ(E) , φ(E) < φ(G); • Let F 6= E 2 Coh(C) be semistable (resp. stable) such that φ(E) > φ(F ) (resp. φ(E) ≥ φ(F )). Then Hom(E; F ) = 0; • If E; F are stable and φ(E) ≥ φ(F ), then either E ' F or Hom(E; F ) = 0; • if E is stable, then End(E) ' k. Proof. For the first assertion: write equivalent inequalities for slopes and use ad- ditivity of degree and rank. The second assertion is proved as follows: let a be a nontrivial map a : E ! F , let K ⊂ E be the kernel. Then by semistability and the first statement: φ(E=K) ≥ φ(E) > φ(F ) ≥ φ(E=K) a contradiction. Similarly one proves the other statements. Proposition 2.5. Every E 2 Coh(C) admits a Harder-Narasimhan filtration 0 = E0 ⊂ E1 ( ::: ( En = E such that the quotients Ai := Ei=Ei−1 are semistable sheaves of phase φ1 > : : : > φn. The Ai are called semistable factors of E, they're unique. Example 2.6. Let's illustrate the above notions on P1. Suppose a coherent sheaf on 1 has the form E = E ⊕ (L O(a )), where E is the torsion part and P 1 a1>:::>an i 1 also the first semistable factor.